#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Maria E Ruwe

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 2 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Donna Odhiambo

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of Cannibalistic and Chaotic Flour Beetles

##### Patrick Nielsen

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.

The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;

The bifurcation diagram for mu adult is on p. 59;

The bifurcation diagram for C pa is on p. 60.

Andy Long

- 2 years 3 weeks ago

#### Clone of Cannibalistic and Chaotic Flour Beetles

##### Lizzy Compton

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.

The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;

The bifurcation diagram for mu adult is on p. 59;

The bifurcation diagram for C pa is on p. 60.

Andy Long

- 2 years 1 month ago

#### Clone of Cannibalistic and Chaotic Flour Beetles

##### Alyssa Farmer

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.

The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;

The bifurcation diagram for mu adult is on p. 59;

The bifurcation diagram for C pa is on p. 60.

Andy Long

- 2 years 1 month ago

#### MAT375 Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")

##### Andrew E Long

**Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")**

Tags: Education, Chaos, Ecology, Biology, PopulationThanks to Insight Author: John Petersen

Edits by Andy Long

Everything that follows the dashes was created by John Petersen (or at least came from his Insight model). I just wanted to make a few comments.

We are looking at Hare and Lynx, of course. Clone this insight, and change the names.

Then read the text below, to get acquainted with one of the most important and well-known examples of a simple system of differential equations in all of mathematics.

http://www.nku.edu/~longa/classes/mat375/mathematica/Lotka-Volterra.nb------------------------------------------------------------

Dynamic simulation modelers are particularly interested in understanding and being able to distinguish between the behavior of stocks and flows that result from internal interactions and those that result from external forces acting on a system.

For some time modelers have been particularly interested in internal interactions that result in stable oscillations in the absence of any external forces acting on a system.

The model in this last scenario was independently developed by Alfred Lotka (1924) and Vito Volterra (1926). Lotka was interested in understanding internal dynamics that might explain oscillations in moth and butterfly populations and the parasitoids that attack them. Volterra was interested in explaining an increase in coastal populations of predatory fish and a decrease in their prey that was observed during World War I when human fishing pressures on the predator species declined.

Both discovered that a relatively simple model is capable of producing the cyclical behaviors they observed.

Since that time, several researchers have been able to reproduce the modeling dynamics in simple experimental systems consisting of only predators and prey. It is now generally recognized that the model world that Lotka and Volterra produced is too simple to explain the complexity of most predator-prey dynamics in nature. And yet, the model significantly advanced our understanding of the critical role of feedback in predator-prey interactions and in feeding relationships that result in community dynamics.

The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.2. The food supply of the predator population depends entirely on the size of the prey population.3. The rate of change of population is proportional to its size.4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.5. Predators have limitless appetite.

As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]

**Prey**

When multiplied out, the prey equation becomesdx/dt = αx - βxy The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.

PredatorsThe predator equation becomes

dy/dt = -

In this equation, {\displaystyle \displaystyle \delta xy} represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). {\displaystyle \displaystyle \gamma y} represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.

Education Chaos Ecology Biology Population Mat375 Lotka Volterra

- 2 months 4 days ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Matthew Gall

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of Cannibalistic and Chaotic Flour Beetles

##### Clay Frink

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.

The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;

The bifurcation diagram for mu adult is on p. 59;

The bifurcation diagram for C pa is on p. 60.

Andy Long

- 2 years 1 month ago

#### Clone of Flakes no more!

##### Terra Ficke

Next up: and SIR, and his interesting model of female birth weights.

- 2 years 2 months ago

#### Clone of Cannibalistic and Chaotic Flour Beetles

##### Austin Hardesty

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.

The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;

The bifurcation diagram for mu adult is on p. 59;

The bifurcation diagram for C pa is on p. 60.

Andy Long

- 2 years 3 weeks ago

#### Clone of Isle Royale: Predator/Prey Model for Moose and Wolves

##### Marosi Balázs

https://insightmaker.com/insight/2068/Isle-Royale-Predator-Prey-Interactions

Thanks Scott Fortmann-Roe.

I've created a Mathematica file that replicates the model, at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker.nb

It allows one to experiment with adjusting the initial number of moose and wolves on the island.

I used steepest descent in Mathematica to optimize the parameters, with my objective data being the ratio of wolves to moose. You can try my (admittedly) kludgy code, at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker-BestFit.nb

{WolfBirthRateFactorStart,

WolfDeathRateStart,

MooseBirthRateStart,

MooseDeathRateFactorStart,

moStart,

woStart} =

{0.000267409,

0.239821,

0.269755,

0.0113679,

591,

23.};

- 1 year 3 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Maria McMahon

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Matthew Gall

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Christopher Milesky

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Maria E Ruwe

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 2 months ago

#### Clone of English Mother/Daughter Birth Weights

##### Parker Kain

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 2 years 1 month ago

#### Clone of English Mother/Daughter Birth Weights

##### Jacob Englert

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....

Andy Long

Next up: an SIR.

- 2 years 2 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Joey Stevens

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of MAT 375 Midterm file: Model of Isle Royale: Predator Prey Interactions

##### Clay Frink

We incorporate logistic growth into the moose dynamics, and we replace the death flow of the moose with a kill rate modeled from the kill rate data found on the Isle Royale website.

Thanks to Jacob Englert for the model if-then-else structure.

I start with these parameters:

Wolf Death Rate = 0.15

Wolf Birth Rate = 0.0187963

Moose Birth Rate = 0.4

Carrying Capacity = 2000

Initial Moose: 563

Initial Wolves: 20

I used RK-4 with step-size 0.1, from 1959 for 60 years.

The moose birth flow is logistic, MBR*M*(1-M/K)

Moose death flow is Kill Rate (in Moose/Year)

Wolf birth flow is WBR*Kill Rate (in Wolves/Year)

Wolf death flow is WDR*W

- 2 years 2 months ago

#### Clone of Flakes no more!

##### Lizzy Compton

Next up: and SIR, and his interesting model of female birth weights.

- 2 years 2 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Allison Zembrodt

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of SIR (poisoning would be better) Markov Model

##### Maria E Ruwe

It's an SIR-type model, but one where the equilibrium (ws,wi,wr) is always the same, even as the weights in the transition matrix change.

Actually it might be better to think of this as a poisoning model: the rate of infection is constant, and independent of the existence of an infected population. That's more like disease due to an environmental effect (e.g. lead-poisoning from smelters, or mercury poisoning from the burning of coal). So infected would mean "effected", and "recovered" might be "treated" -- and ultimately released, to be exposed again.

This shows that the equilibrium does not determine the transition probabilities: two different transition matrices can have the same ultimate equilibrium.

There is a constraint on the infection rate that I haven't figured out how to build in:

InfectionRate < Min[1,wi/ws, wr/ws]

I can allow InfectionRate to vary up to 1 if I take

ws < wi

and

ws < wr

However if you violate that, you'll get interesting solutions with negative values of populations. The dynamics are pretty interesting in that case, however! If you want to see them, you'll have to remove the constraints that I put on the parameters in the Recover and LossOfImmunity parameters.

Thanks Mike! Interesting examples, as always....

Andy Long

- 2 years 1 month ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Rachel Driehaus

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago

#### Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

##### Patrick Nielsen

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at

http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

- 2 years 3 months ago